CN101451889B - Systematic error correction method of interferometer - Google Patents

Systematic error correction method of interferometer Download PDF

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CN101451889B
CN101451889B CN2008101774358A CN200810177435A CN101451889B CN 101451889 B CN101451889 B CN 101451889B CN 2008101774358 A CN2008101774358 A CN 2008101774358A CN 200810177435 A CN200810177435 A CN 200810177435A CN 101451889 B CN101451889 B CN 101451889B
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aberration
function
astigmatic
mrow
coma
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CN101451889A (en
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斋藤隆行
葛宗涛
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Fujinon Corp
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    • G—PHYSICS
    • G11—INFORMATION STORAGE
    • G11B—INFORMATION STORAGE BASED ON RELATIVE MOVEMENT BETWEEN RECORD CARRIER AND TRANSDUCER
    • G11B7/00—Recording or reproducing by optical means, e.g. recording using a thermal beam of optical radiation by modifying optical properties or the physical structure, reproducing using an optical beam at lower power by sensing optical properties; Record carriers therefor
    • G11B7/12—Heads, e.g. forming of the optical beam spot or modulation of the optical beam
    • G11B7/135—Means for guiding the beam from the source to the record carrier or from the record carrier to the detector
    • G11B7/1372—Lenses
    • G11B7/1378—Separate aberration correction lenses; Cylindrical lenses to generate astigmatism; Beam expanders
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01M—TESTING STATIC OR DYNAMIC BALANCE OF MACHINES OR STRUCTURES; TESTING OF STRUCTURES OR APPARATUS, NOT OTHERWISE PROVIDED FOR
    • G01M11/00—Testing of optical apparatus; Testing structures by optical methods not otherwise provided for
    • G01M11/02—Testing optical properties
    • G01M11/0242—Testing optical properties by measuring geometrical properties or aberrations
    • G01M11/0271—Testing optical properties by measuring geometrical properties or aberrations by using interferometric methods
    • G—PHYSICS
    • G11—INFORMATION STORAGE
    • G11B—INFORMATION STORAGE BASED ON RELATIVE MOVEMENT BETWEEN RECORD CARRIER AND TRANSDUCER
    • G11B7/00—Recording or reproducing by optical means, e.g. recording using a thermal beam of optical radiation by modifying optical properties or the physical structure, reproducing using an optical beam at lower power by sensing optical properties; Record carriers therefor
    • G11B7/12—Heads, e.g. forming of the optical beam spot or modulation of the optical beam
    • G11B7/135—Means for guiding the beam from the source to the record carrier or from the record carrier to the detector
    • G11B7/1372—Lenses

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Abstract

The present invention provides a systematic error correction method of an interometer, the method analytically calculates systematic inherent astigmatic aberration component and coma aberration component caused by the systematic errors, thereby correcting detected results of the wave front aberration of lens. Lens (5) to be detected are held at two rotation positions respectively separated by 90DEG relative to a detection optical axis (C), obtained a first and a second aberration functions are classified into aberration function corresponding to Saden aberration, the first and second astigmatic aberration functions corresponding to the astigmatic aberration are calculated. The function added with the first and second astigmatic aberration functions is classified again into aberration function corresponding to Saden aberration, wherein a third astigmatic aberration function corresponding to the astigmatic function is calculated, then the systematic inherent astigmatic aberration component is calculated based on systematic inherent astigmatic aberration function corresponding to onehalf based on the third astigmatic aberration function.

Description

System error correction method for interferometer apparatus
Technical Field
The present invention relates to a method for correcting a system error of an interferometer apparatus for measuring a wavefront aberration of a test lens, and more particularly, to a method for correcting a system error of an interferometer apparatus suitable for separating and obtaining an astigmatic aberration (astigmatic aberration) component and a coma aberration component inherent to a system and an astigmatic aberration component and a coma aberration component inherent to a test lens, which are caused by a system error of the interferometer apparatus.
Background
Conventionally, in order to examine the optical performance of an objective lens for an optical disk such as a CD or a DVD, measurement of a transmitted wavefront (also referred to as a wavefront) by an interferometer is performed. For example, when measuring a test lens designed to convert a plane wave into a spherical wave, the test lens is arranged in a parallel light beam, and an optical test (transmitted wavefront) reflected by a reference spherical surface after passing through the test lens is interfered with a reference light (reference wavefront) from a reference surface, and a wavefront aberration of the test lens is measured from an obtained interference fringe image.
In such transmitted wavefront measurement, since an aberration component caused by a systematic error (a shape error of a reference plane or a reference spherical surface, etc.) of the interferometer device overlaps with a measurement result, it is preferable to suppress the systematic error as much as possible in order to obtain a measurement result with high accuracy, but it is very difficult to completely eliminate the systematic error in terms of processing accuracy.
Conventionally, as a technique for measuring wavefront aberration of a test lens with high accuracy while reducing the influence of such system errors, a method described in patent document 1 below is known. In this method, the lens to be inspected is rotated by a predetermined angle (90 degrees when the astigmatic aberration component is obtained and 180 degrees when the coma aberration component is obtained) around the optical axis, 1 of 2 points of the difference between the measured wavefronts obtained at the 2 positions before and after the rotation is obtained, and the astigmatic aberration component and the coma aberration component unique to the lens to be inspected can be obtained from the astigmatic aberration coefficient and the coma aberration coefficient when the measured wavefronts are expanded by the zernike polynomial.
Patent document 1: japanese patent No. 3230536
With the method described in patent document 1, the influence of the system-specific astigmatism component and coma aberration component can be consequently removed from the obtained measured wavefront data, but no consideration is given to the determination of the system-specific astigmatism component and coma aberration component.
Therefore, in order to accurately obtain the astigmatism component and the coma aberration component inherent to the test lens, it is necessary to perform transmitted wavefront measurement at 2 rotational positions for each test lens to be measured.
On the other hand, if the astigmatic aberration component and the coma aberration component inherent to the system can be obtained, the astigmatic aberration component and the coma aberration component inherent to the test lens can be obtained with high accuracy by correcting the wavefront aberration result obtained by 1 time of transmitted wavefront measurement at an arbitrary rotational position, and therefore, the time required for measurement or calculation can be significantly shortened.
Disclosure of Invention
The present invention has been made in view of the above circumstances, and an object thereof is to provide a systematic error correction method for an interferometer apparatus, which can analytically obtain a astigmatic aberration component and a coma aberration component inherent to a system due to a systematic error of the interferometer apparatus, and correct a measurement result of a wavefront aberration of a test lens.
The present invention provides a systematic error correction method for an interferometer apparatus 1, wherein a wavefront aberration of a lens to be examined is measured from an interference fringe image obtained by interference of light of a transmitted wavefront of the lens to be examined and a reference wavefront, a system-specific astigmatic aberration component caused by a systematic error of the interferometer apparatus is analytically obtained, and a measurement result of the wavefront aberration is corrected,
holding the test lens at an arbitrary rotational position with respect to a measurement optical axis of the interferometer device, and obtaining a 1 st aberration function corresponding to the wavefront aberration at the arbitrary rotational position from the interference fringe image obtained at the arbitrary rotational position;
holding the test lens at a rotational position for comparison rotated by 90 degrees around the measurement optical axis from the arbitrary rotational position, and obtaining a 2 nd aberration function corresponding to the wavefront aberration at the rotational position for comparison from the interference fringe image obtained at the rotational position for comparison;
classifying the 1 st aberration function into aberration functions corresponding to the respective seidel aberrations, and determining an aberration function corresponding to the astigmatic aberration as the 1 st astigmatic aberration function;
classifying the 2 nd aberration function into aberration functions corresponding to the respective seidel aberrations, and determining an aberration function corresponding to the astigmatic aberration as the 2 nd astigmatic aberration function;
reclassifying the function obtained by adding the 1 st astigmatic aberration function and the 2 nd astigmatic aberration function into aberration functions respectively corresponding to the seidel aberrations, and determining an aberration function corresponding to the astigmatic aberration as a 3 rd astigmatic aberration function;
a function of 1 time 2 of the 3 rd astigmatic aberration function is used as a system-specific astigmatic aberration function representing the system-specific astigmatic aberration components, and the system-specific astigmatic aberration components are obtained from the system-specific astigmatic aberration function.
The present invention provides a systematic error correction method for an interferometer apparatus 2, wherein a wavefront aberration of a lens to be inspected is measured from an interference fringe image obtained by interference of light of a transmitted wavefront of the lens to be inspected and a reference wavefront, a coma aberration component inherent to a system due to a systematic error of the interferometer apparatus is analytically obtained, a measurement result of the wavefront aberration is corrected,
holding the test lens at an arbitrary rotational position with respect to a measurement optical axis of the interferometer device, and obtaining a 1 st aberration function corresponding to the wavefront aberration at the arbitrary rotational position from the interference fringe image obtained at the arbitrary rotational position;
holding the test lens at a rotational position for comparison rotated by 180 degrees around the measurement optical axis from the arbitrary rotational position, and obtaining a 2 nd aberration function corresponding to the wavefront aberration at the rotational position for comparison from the interference fringe image obtained at the rotational position for comparison;
classifying the 1 st aberration function into aberration functions corresponding to the seider aberrations respectively, and calculating an aberration function corresponding to the coma aberration as the 1 st coma aberration function;
classifying the 2 nd aberration function into aberration functions corresponding to the seider aberration, and calculating an aberration function corresponding to the coma aberration as the 2 nd coma aberration function;
reclassifying a function obtained by adding the 1 st coma aberration function and the 2 nd coma aberration function into aberration functions respectively corresponding to the seider aberration, and calculating an aberration function corresponding to the coma aberration as a 3 rd coma aberration function;
a function of 1/2 of the 3 rd coma aberration function is used as a system-specific coma aberration function representing the system-specific coma aberration component, and the system-specific coma aberration component is obtained from the system-specific coma aberration function.
A systematic error correction method of a 3 rd interferometer apparatus according to the present invention is characterized in that, in an interferometer apparatus for measuring a wavefront aberration of a lens to be inspected based on an interference fringe image obtained by interference of light of a transmitted wavefront of the lens to be inspected and a reference wavefront, a system-specific astigmatic aberration component and coma aberration component due to a systematic error of the interferometer apparatus are analytically obtained, and a measurement result of the wavefront aberration is corrected; wherein,
holding the test lens at an arbitrary rotational position with respect to a measurement optical axis of the interferometer device, and obtaining a 1 st aberration function corresponding to the wavefront aberration at the arbitrary rotational position from the interference fringe image obtained at the arbitrary rotational position;
holding the test lens at a 1 st rotational position for comparison rotated by 90 degrees around the measurement optical axis from the arbitrary rotational position, and obtaining a 2 nd aberration function corresponding to the wavefront aberration at the 1 st rotational position for comparison from the interference fringe image obtained at the 1 st rotational position for comparison;
holding the test lens at a 2 nd rotational position for comparison rotated by 180 degrees around the measurement optical axis from the arbitrary rotational position, and obtaining a 3 rd aberration function corresponding to the wavefront aberration at the 2 nd rotational position for comparison from the interference fringe image obtained at the 2 nd rotational position for comparison;
classifying the 1 st aberration function into aberration functions respectively corresponding to the seider aberrations, and respectively obtaining an aberration function corresponding to the astigmatic aberration as the 1 st astigmatic aberration function and an aberration function corresponding to the coma aberration as the 1 st coma aberration function;
classifying the 2 nd aberration function into aberration functions corresponding to the respective seidel aberrations, and determining an aberration function corresponding to the astigmatic aberration as the 2 nd astigmatic aberration function;
classifying the 3 rd aberration function into aberration functions corresponding to the seider aberration, and calculating an aberration function corresponding to the coma aberration as a 2 nd coma aberration function;
reclassifying the function obtained by adding the 1 st astigmatic aberration function and the 2 nd astigmatic aberration function into aberration functions respectively corresponding to the seidel aberrations, and determining an aberration function corresponding to the astigmatic aberration as a 3 rd astigmatic aberration function;
reclassifying a function obtained by adding the 1 st coma aberration function and the 2 nd coma aberration function into aberration functions respectively corresponding to the seider aberration, and calculating an aberration function corresponding to the coma aberration as a 3 rd coma aberration function;
determining an astigmatic aberration component inherent to the system from a function of 1/2 of the 3 rd astigmatic aberration function as a system-inherent astigmatic aberration function representing the system-inherent astigmatic aberration component; a function of 1/2 of the 3 rd coma aberration function is used as a system-specific coma aberration function representing the system-specific coma aberration component, and the system-specific coma aberration component is obtained from the system-specific coma aberration function.
In the systematic error correction method of the 1 st interferometer of the present invention, the functions for which the difference between the 1 st astigmatic aberration function and the 2 nd astigmatic aberration function is obtained may be classified into aberration functions corresponding to respective seidel aberrations, and the aberration function corresponding to the astigmatic aberration among them may be obtained as the 4 th astigmatic aberration function; a function of 1 time 2 of the 4 th astigmatic aberration function is used as an astigmatic aberration function specific to the lens to be inspected which represents an astigmatic aberration component specific to the lens to be inspected, and the astigmatic aberration component specific to the lens to be inspected is obtained from the astigmatic aberration function specific to the lens to be inspected.
In the system error correction method of the 2 nd interferometer of the present invention, the functions in which the difference between the 1 st coma aberration function and the 2 nd coma aberration function is obtained may be classified into aberration functions corresponding to respective seidel aberrations, and the aberration function corresponding to the coma aberration among them may be obtained as the 4 th coma aberration function; a function of 1 time 2 of the 4 th coma aberration function is used as a coma aberration function specific to the lens to be detected representing the coma aberration component specific to the lens to be detected, and the coma aberration component specific to the lens to be detected is obtained from the coma aberration function specific to the lens to be detected.
In the systematic error correction method of the 3 rd interferometer apparatus according to the present invention, the functions in which the difference between the 1 st astigmatic aberration function and the 2 nd astigmatic aberration function is obtained may be classified into aberration functions corresponding to respective seidel aberrations, and the aberration function corresponding to the astigmatic aberration among them may be obtained as the 4 th astigmatic aberration function; classifying a function in which a difference between the 1 st coma aberration function and the 2 nd coma aberration function is obtained into aberration functions corresponding to respective seidel aberrations, and determining an aberration function corresponding to a coma aberration as a 4 th coma aberration function;
determining an astigmatic aberration component specific to the lens to be inspected from a function of 1-fold of 2-fold of the 4 th astigmatic aberration function as an astigmatic aberration function specific to the lens to be inspected, the astigmatic aberration function specific to the lens to be inspected representing an astigmatic aberration component specific to the lens to be inspected; a function of 1 time 2 of the 4 th coma aberration function is used as a coma aberration function specific to the lens to be detected, which represents the coma aberration component specific to the lens to be detected, and the coma aberration component specific to the lens to be detected is obtained from the coma aberration function specific to the lens to be detected.
With the above configuration, the systematic error correction method of the interferometer apparatus according to the present invention can analytically determine the astigmatism aberration component and the coma aberration component inherent to the system due to the systematic error of the interferometer apparatus, and can correct the measurement result of the wavefront aberration from the determined astigmatism aberration component and coma aberration component inherent to the system.
The obtained value of the astigmatism aberration component or the coma aberration component inherent to the system needs to be periodically corrected, but can be used for correction of the obtained measurement result of the wavefront aberration in the transmission wavefront measurement with respect to the lens to be inspected a plurality of times.
Therefore, as in the conventional method, it is not necessary to measure transmitted wavefronts at 2 different rotational positions for each test lens as an assumed object, and it is possible to accurately obtain the astigmatic aberration component and coma aberration component specific to the test lens.
Drawings
Fig. 1 is a schematic configuration diagram of an interferometer to which an embodiment of the present invention is applied.
In the figure: 1-interferometer device, 5-lens to be inspected, 10-optical interference part, 11-light source part, 12-lens for enlarging beam diameter, 13-beam splitter, 13 a-beam splitting surface, 14-collimating lens; 15-reference plate, 15 a-reference plane, 16-placing table, 17-spherical mirror, 17 a-reference spherical surface, 20-image pick-up unit, 21-imaging lens, 22-camera, 30-analysis unit, 31-analysis unit, 32-input unit, 33-image display unit, C-measured optical axis
Detailed Description
Hereinafter, an embodiment of a systematic error correction method of an interferometer according to the present invention will be described in detail with reference to the drawings. Fig. 1 is a schematic configuration diagram of an interferometer to which an embodiment of the present invention is applied.
The interferometer 1 shown in fig. 1 is a fizeau type interferometer for measuring a transmitted wavefront of a test lens 5 designed to convert a plane wave into a spherical wave, and includes an optical interference unit 10, an imaging unit 20, and an analysis unit 30.
The optical interference unit 10 includes a light source unit 11 including a laser light source or the like, a beam diameter enlarging lens 12, a beam splitter 13 having a beam splitting surface 13a, a collimator lens 14 for collimating a divergent beam, a transmissive reference plate 15 having a reference plane 15a, a mount 16 for holding the test lens 5, and a spherical mirror 17 having a reference spherical surface 17a for reflecting a transmitted wavefront from the test lens 5 backward, and interference light of the transmitted wavefront traveling downward in the figure through the transmissive reference plate 15 and the test lens 5, reflected by the reference plane 17a, and again transmitted through the test lens 5 and the reference wavefront traveling downward in the figure reflected by the reference plane 15a enters the imaging unit 20 through the collimator lens 14 and the beam splitter 13.
The mount table 16 is provided with a 2-axis tilt adjustment table for adjusting the tilt of the test lens 5 with respect to the measurement optical axis C of the interferometer apparatus 1, a 3-axis position adjustment table for adjusting the positions of the test lens 5 with respect to the measurement optical axis C of the interferometer apparatus 1 and the reference spherical surface 17a, and a rotary table (not shown) for adjusting the rotational position of the test lens 5 with respect to the measurement optical axis C of the interferometer apparatus 1. Further, the reference plate 15 is configured to: the position of the reference surface 15a on the optical path can be changed slightly by being supported by a not-shown fringe scanning adapter (フリンジスキャンアダプタ) and being moved slightly in the direction of the optical axis C of the interferometer 1 by the fringe scanning adapter.
The imaging unit 20 includes an imaging lens 21 and a camera 22, captures interference light incident from the optical interference unit 10, captures an interference fringe image having transmitted wavefront information of the test lens 5, and outputs the interference fringe image to the analysis unit 30.
The analysis unit 30 includes an analysis device 31 that analyzes the interference fringe image input from the imaging unit 20, an image display device 32 that displays the captured interference fringe image, the analysis result, and the like, and an input device 33 that performs various inputs to the analysis device 31. The analysis device 31 is configured by a computer or the like, and includes a storage unit such as a hard disk in which various programs are stored, a CPU that performs various arithmetic processes, and the like.
Hereinafter, a procedure of a systematic error correction method (hereinafter, may be referred to as "the method of the present embodiment") of an interferometer according to an embodiment of the present invention will be described. The method of the present embodiment is applied to the interferometer 1 described above, and analytically obtains astigmatic aberration (also referred to as astigmatism) and coma aberration (also referred to as coma aberration) due to a system error of the interferometer 1, such as a shape error of the reference plane 15a or the reference spherical surface 17a of the interferometer 1, and corrects the measurement result of the wavefront aberration of the test lens 5 obtained by the interferometer 1. The arithmetic processing in the following steps can be performed by the analysis device 31. Further, calibration adjustment of the interferometer device 1 or inclination or position adjustment of the object 5 is completed.
(1) As shown in fig. 1, the test lens 5 is held at an arbitrary rotational position with respect to the measurement optical axis C of the interferometer apparatus 1, and the 1 st aberration function corresponding to the wavefront aberration of the test lens 5 at the arbitrary rotational position is obtained from the interference fringe image obtained at the arbitrary rotational position.
(2) The test lens 5 is held at the 1 st rotational position for comparison rotated by 90 degrees around the measurement optical axis C from the arbitrary rotational position, and the 2 nd aberration function corresponding to the wavefront aberration of the test lens 5 at the 1 st rotational position for comparison is obtained from the interference fringe image obtained at the 1 st rotational position for comparison.
(3) The test lens 5 is held at the 2 nd rotational position for comparison rotated by 180 degrees around the measurement optical axis from the arbitrary rotational position, and the 3 rd aberration function corresponding to the wavefront aberration of the test lens 5 at the 2 nd rotational position for comparison is obtained from the interference fringe image obtained at the 2 nd rotational position for comparison.
The 1 st to 3 rd aberration functions may be obtained by, for example, obtaining the wavefront shape by a phase shift method and developing the wavefront shape into zernike polynomials (zernike polynomials of 4 th, 6 th, 8 th, 10 th, etc.) (the same applies to other aberration functions below).
(4) The 1 st aberration function is classified into aberration functions corresponding to seidel aberrations (for example, tilt (distortion aberration) as 3-order aberration, magnification (defocus, field curvature), astigmatism (astigmatism), coma (coma), and spherical aberration (spherical aberration), which are the same below), and the aberration function corresponding to the astigmatism aberration is determined as the 1 st astigmatism difference function and the aberration function corresponding to the coma aberration is determined as the 1 st coma function.
(5) The 2 nd aberration function is classified into aberration functions corresponding to the respective seidel aberrations, and the aberration function corresponding to the astigmatic aberration is determined as the 2 nd astigmatic aberration function.
(6) The 3 rd aberration function is classified into aberration functions corresponding to the respective seidel aberrations, and an aberration function corresponding to the coma aberration among them is obtained as the 2 nd coma aberration function.
(7) The function obtained by adding the 1 st astigmatic aberration function and the 2 nd astigmatic aberration function is further classified into aberration functions corresponding to the respective seidel aberrations, and the aberration function corresponding to the astigmatic aberration is obtained as the 3 rd astigmatic aberration function.
(8) The function of adding the 1 st coma aberration function and the 2 nd coma aberration function is reclassified as aberration functions corresponding to the respective seider aberrations, and the aberration function corresponding to the coma aberration among them is found as the 3 rd coma aberration function.
(9) A function of 1 time 2 of the 3 rd astigmatic aberration function is used as a system-specific astigmatic aberration function representing a system-specific astigmatic aberration component of the interferometer device 1, and the system-specific astigmatic aberration component is obtained from the system-specific astigmatic aberration function.
(10) A function of 1/2 of the 3 rd coma aberration function is set as a system-specific coma aberration function indicating a system-specific coma aberration component of the interferometer 1, and the system-specific coma aberration component is obtained from the system-specific coma aberration function.
(11) Classifying a function of a difference between the 1 st astigmatic aberration function and the 2 nd astigmatic aberration function into aberration functions corresponding to respective seidel aberrations, and determining an aberration function corresponding to the astigmatic aberration as a 4 th astigmatic aberration function; classifying a function of a difference between the 1 st coma aberration function and the 2 nd coma aberration function into aberration functions corresponding to respective seider aberrations, and determining an aberration function corresponding to the coma aberration as the 4 th coma aberration function
(12) A function of 1 time 2 of the 4 th astigmatic aberration function is used as an astigmatic aberration function specific to the lens to be inspected which represents an astigmatic aberration component specific to the lens to be inspected 5, and the astigmatic aberration component specific to the lens to be inspected 5 is obtained from the astigmatic aberration function specific to the lens to be inspected; a function of 1 times 2 of the 4 th coma aberration function is set as a coma aberration function specific to the lens to be inspected which represents a coma aberration component specific to the lens to be inspected 5, and the coma aberration component specific to the lens to be inspected is obtained from the aberration function specific to the lens to be inspected.
The operation of the method of the present embodiment is verified by using a mathematical expression.
If an aberration function related to the measured value of the wavefront aberration of the interferometer 1 is set as Mψ(ρ, θ) and an aberration function W relating to wavefront aberration components specific to the subject 5ψ(ρ, θ) and an aberration function S of a wavefront aberration component inherent to the system of the interferometer 1ψ(ρ, θ) has the following relationship (1). The suffix ψ denotes a rotational position angle of the test lens 5 with respect to the measurement optical axis C of the interferometer apparatus 1 (for example, ψ is 0 °, 90 °, 180 °, 270 °, or the like), and the parameters ρ and θ denote a distance and an angle from the center of the polar coordinate. In the following, the aberration function M is described aboveψIn (ρ, θ), when ψ is 90 ° and 180 °, the aberration functions are treated to correspond to the 1 st, 2 nd, and 3 rd aberration functions described above, respectively.
[ mathematical formula 1]
Mψ(ρ，θ)＝Wψ(ρ，θ)+Sψ(ρ，θ) …… (1)
Further, the aberration function M is setψ(ρ，θ)、Wψ(rho, theta) and Sψ(ρ, θ) are classified into respective Saider aberrations, and an aberration function corresponding to the astigmatic aberration is defined as Aam(ρ，θ)、Aaw(ρ, θ) and Aas(ρ, θ), these are generally expressed by the following formulas (2) to (4).
[ mathematical formula 2]
Aam(ρ，θ)＝Aamρ2cos2(θ-φam) …… (2)
Aaw(ρ，θ)＝Aawρ2cos2(θ-φaw) …… (3)
Aas(ρ，θ)＝Aasρ2cos2(θ-φas) …… (4)
Here, phiam、φaw、φasIs an angle (non-point angle) indicating the direction of each astigmatic aberration.
The following formulas (5) and (6) are based on the relationship of the above formula (1)) The relationship of (1) holds. The value in parentheses of the suffix is the value of ψ (the same applies hereinafter). Furthermore, Aam(0) (ρ, θ) and Aam(90)(ρ, θ) corresponds to the 1 st astigmatic aberration function and the 2 nd astigmatic aberration function, respectively.
[ mathematical formula 3]
Aam(0)(ρ，θ)＝Aaw(0)(ρ，θ)+Aas(ρ，θ) …… (5)
Aam(90)(ρ，θ)＝Aaw(0)(ρ，θ+90)+Aas(ρ，θ) …… (6)
The following expressions (7) and (8) are obtained by adding the above expressions (5) and (6) and performing the arithmetic processing of the above expressions (2) to (4). The following expression (7) corresponds to a function of adding the 1 st astigmatic aberration function and the 2 nd astigmatic aberration function, and the following expression (8) corresponds to a function of subtracting the 2 nd astigmatic aberration function from the 1 st astigmatic aberration function.
[ mathematical formula 4]
Aam(0)(ρ，θ)+Aam(90)(ρ，θ)＝Aaw(0)ρ2cos2(θ-φaw(0))+Aaw(0)ρ2cos2(θ-φaw(0)+90)+2Aas(ρ，θ)
＝2Aas(ρ，θ)+Aaw(0)ρ2 …… (7)
Aam(0)(ρ，θ)-Aam(90)(ρ，θ)＝Aaw(0)ρ2cos2(θ-φaw(0))-Aaw(0)ρ2cos2(θ-φaw(0)+90)
＝2Aaw(0)(ρ，θ)-Aaw(0)ρ2 …… (8)
At the upper partIn the formulae (7), (8) < A >aw(0)ρ2Showing the magnification aberration component in the seidel aberration. Therefore, the aberration functions expressed by the above-described equations (7) and (8) are classified into aberration functions corresponding to the seidel aberrations, and the aberration function corresponding to the astigmatic aberration among them is determined ({ }sIndicating the case where this operation is performed), the magnification aberration component is removed, and therefore the relationships of the following expressions (9) and (10) are established. The left side of the following expression (9) and the left side of the following expression (10) correspond to the 3 rd astigmatic aberration function and the 4 th astigmatic aberration function.
[ math figure 5]
{Aam(0)(ρ，θ)+Aam(90)(ρ，θ)}s＝2Aas(ρ，θ) …… (9)
{Aam(0)(ρ，θ)-Aam(90)(ρ，θ)}s＝2Aaw(0)(ρ，θ) …… (10)
As shown in the above equation (9), the function of 1/2 of the 3 rd astigmatic aberration function becomes a system-specific astigmatic aberration function representing the system-specific astigmatic aberration components of the interferometer apparatus 1, and the system-specific astigmatic aberration components are obtained from the system-specific astigmatic aberration function.
As shown in the above equation (10), the function of 1 × 2 of the 4 th astigmatic aberration function is an astigmatic aberration function specific to the test lens 5 indicating an astigmatic aberration component specific to the test lens 5, and the astigmatic aberration component specific to the test lens 5 is obtained from the astigmatic aberration function specific to the test lens.
Similarly, the relationship of the following expressions (11) and (12) is derived for the coma aberration. The left side of the following expression (11) and the left side of the following expression (12) correspond to the 3 rd coma aberration function and the 4 th coma aberration function. In addition, the following { } c denotes an operation of obtaining an aberration function corresponding to coma aberration.
[ mathematical formula 6]
{Acm(0)(ρ，θ)+Acm(180)(ρ，θ)}c＝2Acs(ρ，θ) …… (11)
{Acm(0)(ρ，θ)-Acm(180)(ρ，θ)}c＝2Acs(0)(ρ，θ) …… (12)
The principle that the relationship of the following expression (13) holds is generally used.
[ math figure 7]
<math> <mrow> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msub> <mi>A</mi> <mi>cm</mi> </msub> <mrow> <mo>(</mo> <mi>&rho;</mi> <mo>,</mo> <mi>&theta;</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>A</mi> <mi>cm</mi> </msub> <msup> <mi>&rho;</mi> <mn>3</mn> </msup> <mi>cos</mi> <mrow> <mo>(</mo> <mi>&theta;</mi> <mo>-</mo> <msub> <mi>&phi;</mi> <mi>cm</mi> </msub> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>A</mi> <mi>cw</mi> </msub> <mrow> <mo>(</mo> <mi>&rho;</mi> <mo>,</mo> <mi>&theta;</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>A</mi> <mi>cw</mi> </msub> <msup> <mi>&rho;</mi> <mn>3</mn> </msup> <mi>cos</mi> <mo>(</mo> <mi>&theta;</mi> <mo>-</mo> <msub> <mi>&phi;</mi> <mi>cw</mi> </msub> <mo>)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>A</mi> <mi>cs</mi> </msub> <mrow> <mo>(</mo> <mi>&rho;</mi> <mo>,</mo> <mi>&theta;</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>A</mi> <mi>cs</mi> </msub> <msup> <mi>&rho;</mi> <mn>3</mn> </msup> <mi>cos</mi> <mo>(</mo> <mi>&theta;</mi> <mo>-</mo> <msub> <mi>&phi;</mi> <mi>cs</mi> </msub> <mo>)</mo> </mtd> </mtr> </mtable> </mfenced> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> </mrow></math>
Here, A iscm(ρ，θ)、Acw(ρ, θ) and Acs(ρ, θ) is expressed in relation to the above-mentioned aberration function Mψ(ρ，θ)、Wψ(rho, theta) and Sψ(rho, theta) are classified into aberration functions, phi, corresponding to coma aberration for each Saider aberrationcm、φcw、φcsIs an angle (coma angle) indicating the direction of each coma aberration.
As shown by the above equation (11), the function of 1/2 of the above-mentioned 3 rd coma aberration function becomes a system-specific coma aberration function indicating a system-specific coma aberration component of the interferometer 1, and the system-specific coma aberration component is obtained from the system-specific coma aberration function.
In addition, as shown by the above equation (12), a function of 1 times 2 of the 4 th coma aberration function is used as a coma aberration function specific to the subject lens 5, which represents a coma aberration component specific to the subject lens 5, and the coma aberration component specific to the subject lens 5 is obtained from the coma aberration function specific to the subject lens.
Then, when the astigmatic aberration component inherent in the system is obtained, first, the relationship of the following expression (14) is established based on the relationship of the above expression 9
[ mathematical formula 8]
<math> <mrow> <msub> <mi>A</mi> <mi>cm</mi> </msub> <mrow> <mo>(</mo> <mi>&rho;</mi> <mo>,</mo> <mi>&theta;</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <msub> <mi>A</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </msub> <mrow> <mo>(</mo> <mi>&rho;</mi> <mo>,</mo> <mi>&theta;</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>A</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>90</mn> <mo>)</mo> </mrow> </mrow> </msub> <mrow> <mo>(</mo> <mi>&rho;</mi> <mo>,</mo> <mi>&theta;</mi> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow></math>
<math> <mrow> <mo>=</mo> <mfrac> <mrow> <msub> <mi>A</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>A</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>90</mn> <mo>)</mo> </mrow> </mrow> </msub> </mrow> <mn>4</mn> </mfrac> <msup> <mi>&rho;</mi> <mn>2</mn> </msup> <mo>+</mo> <mfrac> <mrow> <msub> <mi>A</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </msub> <msup> <mi>&rho;</mi> <mn>2</mn> </msup> <mi>cos</mi> <mn>2</mn> <mrow> <mo>(</mo> <mi>&theta;</mi> <mo>-</mo> <msub> <mi>&phi;</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>A</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>90</mn> <mo>)</mo> </mrow> </mrow> </msub> <msup> <mi>&rho;</mi> <mn>2</mn> </msup> <mi>cos</mi> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mo>(</mo> <mi>&theta;</mi> <mo>-</mo> <msub> <mi>&phi;</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mi>&phi;</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>-</mo> <msub> <mi>&phi;</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>90</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> <mn>4</mn> </mfrac> </mrow></math>
……(14)
Line 2, item 1 of equation (14) above represents the magnification aberration component, and is removed in equation (9), so equation (15) holds.
[ mathematical formula 9]
<math> <mrow> <msub> <mi>A</mi> <mi>cs</mi> </msub> <mrow> <mo>(</mo> <mi>&rho;</mi> <mo>,</mo> <mi>&theta;</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <msqrt> <msup> <mi>M</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>N</mi> <mn>2</mn> </msup> </msqrt> <msup> <mi>cos</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mi>&theta;</mi> <mo>-</mo> <msub> <mi>&phi;</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>&alpha;</mi> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </mfrac> <msup> <mi>&rho;</mi> <mn>2</mn> </msup> <mo>+</mo> <mfrac> <msqrt> <msup> <mi>M</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>N</mi> <mn>2</mn> </msup> </msqrt> <mn>2</mn> </mfrac> <msup> <mi>&rho;</mi> <mn>2</mn> </msup> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> <mrow> <mo>(</mo> <mn>15</mn> <mo>)</mo> </mrow> </mrow></math>
Here, M, N and α are represented by the following formula (16).
[ mathematical formula 10]
<math> <mrow> <mfenced open='{' close='' separators=' ,'> <mtable> <mtr> <mtd> <mi>M</mi> <mo>=</mo> <msub> <mi>A</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>A</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>90</mn> <mo>)</mo> </mrow> </mrow> </msub> <mi>cos</mi> <mn>2</mn> <mrow> <mo>(</mo> <msub> <mi>&phi;</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>-</mo> <msub> <mi>&phi;</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>90</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>N</mi> <mo>=</mo> <msub> <mi>A</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>90</mn> <mo>)</mo> </mrow> </mrow> </msub> <mi>sin</mi> <mn>2</mn> <mrow> <mo>(</mo> <msub> <mi>&phi;</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>-</mo> <msub> <mi>&phi;</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>90</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>&alpha;</mi> <mo>=</mo> <mi>arctan</mi> <mrow> <mo>(</mo> <mfrac> <mi>N</mi> <mi>M</mi> </mfrac> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> <mrow> <mo>(</mo> <mn>16</mn> <mo>)</mo> </mrow> </mrow></math>
Similarly, when the astigmatic aberration component specific to the subject is obtained, the relationship of the following expression (17) is established from the relationship of the above expression (10).
[ mathematical formula 11]
<math> <mrow> <msub> <mi>A</mi> <mi>cw</mi> </msub> <mrow> <mo>(</mo> <mi>&rho;</mi> <mo>,</mo> <mi>&theta;</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <msub> <mi>A</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </msub> <mrow> <mo>(</mo> <mi>&rho;</mi> <mo>,</mo> <mi>&theta;</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>A</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>90</mn> <mo>)</mo> </mrow> </mrow> </msub> <mrow> <mo>(</mo> <mi>&rho;</mi> <mo>,</mo> <mi>&theta;</mi> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow></math>
<math> <mrow> <mo>=</mo> <mfrac> <mrow> <msub> <mi>A</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>-</mo> <msub> <mi>A</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>90</mn> <mo>)</mo> </mrow> </mrow> </msub> </mrow> <mn>4</mn> </mfrac> <msup> <mi>&rho;</mi> <mn>2</mn> </msup> <mo>+</mo> <mfrac> <mrow> <msub> <mi>A</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </msub> <msup> <mi>&rho;</mi> <mn>2</mn> </msup> <mi>cos</mi> <mn>2</mn> <mrow> <mo>(</mo> <mi>&theta;</mi> <mo>-</mo> <msub> <mi>&phi;</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>A</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>90</mn> <mo>)</mo> </mrow> </mrow> </msub> <msup> <mi>&rho;</mi> <mn>2</mn> </msup> <mi>cos</mi> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mo>(</mo> <mi>&theta;</mi> <mo>-</mo> <msub> <mi>&phi;</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mi>&phi;</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>-</mo> <msub> <mi>&phi;</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>90</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> <mn>4</mn> </mfrac> </mrow></math>
……(17)
Line 2, item 1 of equation (17) represents the magnification aberration component, and is removed from equation (10), so equation (18) holds.
[ mathematical formula 12]
<math> <mrow> <msub> <mi>A</mi> <mi>cw</mi> </msub> <mrow> <mo>(</mo> <mi>&rho;</mi> <mo>,</mo> <mi>&theta;</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <msqrt> <msup> <mi>K</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>N</mi> <mn>2</mn> </msup> </msqrt> <msup> <mi>cos</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mi>&theta;</mi> <mo>-</mo> <msub> <mi>&phi;</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>&beta;</mi> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </mfrac> <msup> <mi>&rho;</mi> <mn>2</mn> </msup> <mo>+</mo> <mfrac> <msqrt> <msup> <mi>K</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>N</mi> <mn>2</mn> </msup> </msqrt> <mn>2</mn> </mfrac> <msup> <mi>&rho;</mi> <mn>2</mn> </msup> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> <mrow> <mo>(</mo> <mn>18</mn> <mo>)</mo> </mrow> </mrow></math>
Here, K and β are represented by the following formula (19).
[ mathematical formula 13]
<math> <mrow> <mfenced open='{' close='' separators=' ,'> <mtable> <mtr> <mtd> <mi>K</mi> <mo>=</mo> <msub> <mi>A</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>-</mo> <msub> <mi>A</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>90</mn> <mo>)</mo> </mrow> </mrow> </msub> <mi>cos</mi> <mn>2</mn> <mrow> <mo>(</mo> <msub> <mi>&phi;</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>-</mo> <msub> <mi>&phi;</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>90</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>&beta;</mi> <mo>=</mo> <mi>arctan</mi> <mrow> <mo>(</mo> <mfrac> <mi>N</mi> <mi>K</mi> </mfrac> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> <mrow> <mo>(</mo> <mn>19</mn> <mo>)</mo> </mrow> </mrow></math>
When the coma aberration component inherent to the system is obtained, first, the relationship of the following expression (20) is established from the relationship of the above expression (11).
[ mathematical formula 14]
<math> <mrow> <msub> <mi>A</mi> <mi>cs</mi> </msub> <mrow> <mo>(</mo> <mi>&rho;</mi> <mo>,</mo> <mi>&theta;</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <msub> <mi>A</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </msub> <mrow> <mo>(</mo> <mi>&rho;</mi> <mo>,</mo> <mi>&theta;</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>A</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>180</mn> <mo>)</mo> </mrow> </mrow> </msub> <mrow> <mo>(</mo> <mi>&rho;</mi> <mo>,</mo> <mi>&theta;</mi> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow></math>
……(20)
<math> <mrow> <mo>=</mo> <mfrac> <mrow> <msqrt> <msup> <mi>P</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>Q</mi> <mn>2</mn> </msup> </msqrt> <mi>cos</mi> <mrow> <mo>(</mo> <mi>&theta;</mi> <mo>-</mo> <msub> <mi>&phi;</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>+</mo> <mi>&gamma;</mi> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </mfrac> <msup> <mi>&rho;</mi> <mn>3</mn> </msup> </mrow></math>
Here, P, Q and γ are expressed by the following formula (21).
[ mathematical formula 15]
<math> <mrow> <mfenced open='{' close='' separators=' ,'> <mtable> <mtr> <mtd> <mi>P</mi> <mo>=</mo> <msub> <mi>A</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>A</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>180</mn> <mo>)</mo> </mrow> </mrow> </msub> <mi>cos</mi> <mrow> <mo>(</mo> <msub> <mi>&phi;</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>-</mo> <msub> <mi>&phi;</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>180</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>Q</mi> <mo>=</mo> <msub> <mi>A</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>180</mn> <mo>)</mo> </mrow> </mrow> </msub> <mi>sin</mi> <mrow> <mo>(</mo> <msub> <mi>&phi;</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>-</mo> <msub> <mi>&phi;</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>180</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>&gamma;</mi> <mo>=</mo> <mi>arctan</mi> <mrow> <mo>(</mo> <mfrac> <mi>Q</mi> <mi>P</mi> </mfrac> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> <mrow> <mo>(</mo> <mn>21</mn> <mo>)</mo> </mrow> </mrow></math>
Similarly, when the coma aberration component inherent to the subject is obtained, the relationship of the following expression (22) is established from the relationship of the above expression (12).
[ mathematical formula 16]
<math> <mrow> <msub> <mi>A</mi> <mi>cw</mi> </msub> <mrow> <mo>(</mo> <mi>&rho;</mi> <mo>,</mo> <mi>&theta;</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <msub> <mi>A</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </msub> <mrow> <mo>(</mo> <mi>&rho;</mi> <mo>,</mo> <mi>&theta;</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>A</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>180</mn> <mo>)</mo> </mrow> </mrow> </msub> <mrow> <mo>(</mo> <mi>&rho;</mi> <mo>,</mo> <mi>&theta;</mi> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow></math>
……(22)
<math> <mrow> <mo>=</mo> <mfrac> <mrow> <msqrt> <msup> <mi>T</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>Q</mi> <mn>2</mn> </msup> </msqrt> <mi>cos</mi> <mrow> <mo>(</mo> <mi>&theta;</mi> <mo>-</mo> <msub> <mi>&phi;</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>-</mo> <mi>&epsiv;</mi> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </mfrac> <msup> <mi>&rho;</mi> <mn>3</mn> </msup> </mrow></math>
Here, T and ∈ are expressed by the following formula (23).
[ mathematical formula 17]
<math> <mrow> <mfenced open='{' close='' separators=' ,'> <mtable> <mtr> <mtd> <mi>T</mi> <mo>=</mo> <msub> <mi>A</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>-</mo> <msub> <mi>A</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>180</mn> <mo>)</mo> </mrow> </mrow> </msub> <mi>cos</mi> <mrow> <mo>(</mo> <msub> <mi>&phi;</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>-</mo> <msub> <mi>&phi;</mi> <mrow> <mi>cm</mi> <mrow> <mo>(</mo> <mn>180</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>&epsiv;</mi> <mo>=</mo> <mi>arctan</mi> <mrow> <mo>(</mo> <mfrac> <mi>Q</mi> <mi>T</mi> </mfrac> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> <mrow> <mo>(</mo> <mn>23</mn> <mo>)</mo> </mrow> </mrow></math>
Although one embodiment of the present invention has been described above, the present invention is not limited to the above embodiment, and various modifications are possible.
For example, in the above-described embodiment, the astigmatism aberration component and the coma aberration component specific to the subject are also analytically obtained by predetermined arithmetic processing together with the astigmatism aberration component and the coma aberration component specific to the system, but the astigmatism aberration component and the coma aberration component specific to the subject may be obtained by subtracting the obtained astigmatism aberration component and coma aberration component specific to the system from the measurement values to correct the measurement values.
In the above-described embodiment, the subject 5 is measured at predetermined 2 rotational positions (for example, 0 ° and 90 ° in the case of astigmatic aberration), and the astigmatic aberration component unique to the system is obtained based on the 2 measurement results, but a plurality of combinations of the predetermined 2 rotational positions are set (for example, combinations of 0 ° and 90 °, 90 ° and 180 °, 270 °, 0 ° (360 °), and the like in the case of astigmatic aberration), and the values of the astigmatic aberration component unique to the system in each combination are obtained from the measurement results of the 2 rotational positions of each combination, and the average value of these values may be set as the value of the astigmatic aberration component unique to the system. This is also the same for the case of obtaining a coma aberration component unique to the system (in the case of coma aberration, for example, a combination of 0 ° and 180 °, 90 ° and 270 °, and the like is used).
Further, the above-described embodiment shows an application example of the case where the transmission wavefront measurement of the inspection lens 5 of the infinity system is performed by the fizeau-type interferometer apparatus 1, but the present invention is not limited to the correspondence of the interferometer apparatus used or the type of the inspection lens. For example, the present invention can be applied to the measurement of a finite system test lens by an interferometer device of another type (for example, a michelson type or a mach-zehnder type).

Claims (6)

1. A systematic error correction method for an interferometer apparatus for measuring a wavefront aberration of a lens to be examined from an interference fringe image obtained by interference of light of a transmitted wavefront of the lens to be examined and a reference wavefront, wherein a system-specific astigmatic aberration component caused by a systematic error of the interferometer apparatus is analytically obtained to correct a measurement result of the wavefront aberration,
holding the test lens at an arbitrary rotational position with respect to a measurement optical axis of the interferometer device, and obtaining a 1 st aberration function corresponding to the wavefront aberration at the arbitrary rotational position from the interference fringe image obtained at the arbitrary rotational position;
holding the test lens at a rotational position for comparison rotated by 90 degrees around the measurement optical axis from the arbitrary rotational position, and obtaining a 2 nd aberration function corresponding to the wavefront aberration at the rotational position for comparison from the interference fringe image obtained at the rotational position for comparison;
classifying the 1 st aberration function into aberration functions corresponding to the respective seidel aberrations, and determining an aberration function corresponding to the astigmatic aberration as the 1 st astigmatic aberration function;
classifying the 2 nd aberration function into aberration functions corresponding to the respective seidel aberrations, and determining an aberration function corresponding to the astigmatic aberration as the 2 nd astigmatic aberration function;
reclassifying the function obtained by adding the 1 st astigmatic aberration function and the 2 nd astigmatic aberration function into aberration functions respectively corresponding to the seidel aberrations, and determining an aberration function corresponding to the astigmatic aberration as a 3 rd astigmatic aberration function;
a function of 1 time 2 of the 3 rd astigmatic aberration function is used as a system-specific astigmatic aberration function representing the system-specific astigmatic aberration components, and the system-specific astigmatic aberration components are obtained from the system-specific astigmatic aberration function.
2. A system error correction method for an interferometer apparatus for measuring a wavefront aberration of a lens to be inspected based on an interference fringe image obtained by interference of light of a transmitted wavefront of the lens to be inspected and a reference wavefront, wherein a coma aberration component inherent to a system due to a system error of the interferometer apparatus is analytically obtained, and a measurement result of the wavefront aberration is corrected,
holding the test lens at an arbitrary rotational position with respect to a measurement optical axis of the interferometer device, and obtaining a 1 st aberration function corresponding to the wavefront aberration at the arbitrary rotational position from the interference fringe image obtained at the arbitrary rotational position;
holding the test lens at a rotational position for comparison rotated by 180 degrees around the measurement optical axis from the arbitrary rotational position, and obtaining a 2 nd aberration function corresponding to the wavefront aberration at the rotational position for comparison from the interference fringe image obtained at the rotational position for comparison;
classifying the 1 st aberration function into aberration functions corresponding to the seider aberrations respectively, and calculating an aberration function corresponding to the coma aberration as the 1 st coma aberration function;
classifying the 2 nd aberration function into aberration functions corresponding to the seider aberration, and calculating an aberration function corresponding to the coma aberration as the 2 nd coma aberration function;
reclassifying a function obtained by adding the 1 st coma aberration function and the 2 nd coma aberration function into aberration functions respectively corresponding to the seider aberration, and calculating an aberration function corresponding to the coma aberration as a 3 rd coma aberration function;
a function of 1/2 of the 3 rd coma aberration function is used as a system-specific coma aberration function representing the system-specific coma aberration component, and the system-specific coma aberration component is obtained from the system-specific coma aberration function.
3. A systematic error correction method of an interferometer apparatus, characterized in that, in an interferometer apparatus for measuring a wavefront aberration of a lens to be tested based on an interference fringe image obtained by interference of light of a transmitted wavefront of the lens to be tested and a reference wavefront, a system-specific astigmatic aberration component and coma aberration component due to a systematic error of the interferometer apparatus are analytically obtained, and a measurement result of the wavefront aberration is corrected; wherein,
holding the test lens at an arbitrary rotational position with respect to a measurement optical axis of the interferometer device, and obtaining a 1 st aberration function corresponding to the wavefront aberration at the arbitrary rotational position from the interference fringe image obtained at the arbitrary rotational position;
holding the test lens at a 1 st rotational position for comparison rotated by 90 degrees around the measurement optical axis from the arbitrary rotational position, and obtaining a 2 nd aberration function corresponding to the wavefront aberration at the rotational position for comparison from the interference fringe image obtained at the 1 st rotational position for comparison;
holding the test lens at a 2 nd rotational position for comparison rotated by 180 degrees around the measurement optical axis from the arbitrary rotational position, and obtaining a 3 rd aberration function corresponding to the wavefront aberration at the 2 nd rotational position for comparison from the interference fringe image obtained at the 2 nd rotational position for comparison;
classifying the 1 st aberration function into aberration functions corresponding to the respective seidel aberrations, and determining an aberration function corresponding to the astigmatic aberration as the 1 st astigmatic aberration function and an aberration function corresponding to the coma aberration as the 1 st coma aberration function;
classifying the 2 nd aberration function into aberration functions corresponding to the respective seidel aberrations, and determining an aberration function corresponding to the astigmatic aberration as the 2 nd astigmatic aberration function;
classifying the 3 rd aberration function into aberration functions corresponding to the seider aberration, and calculating an aberration function corresponding to the coma aberration as a 2 nd coma aberration function;
reclassifying the function obtained by adding the 1 st astigmatic aberration function and the 2 nd astigmatic aberration function into aberration functions respectively corresponding to the seidel aberrations, and determining an aberration function corresponding to the astigmatic aberration as a 3 rd astigmatic aberration function;
reclassifying a function obtained by adding the 1 st coma aberration function and the 2 nd coma aberration function into aberration functions respectively corresponding to the seider aberration, and calculating an aberration function corresponding to the coma aberration as a 3 rd coma aberration function;
determining an astigmatic aberration component inherent to the system from a function of 1/2 of the 3 rd astigmatic aberration function as a system-inherent astigmatic aberration function representing the system-inherent astigmatic aberration component; a function of 1/2 of the 3 rd coma aberration function is used as a system-specific coma aberration function representing the system-specific coma aberration component, and the system-specific coma aberration component is obtained from the system-specific coma aberration function.
4. The systematic error correction method of interferometer apparatus of claim 1,
classifying a function in which a difference between the 1 st astigmatic aberration function and the 2 nd astigmatic aberration function is obtained into aberration functions corresponding to respective seidel aberrations, and determining an aberration function corresponding to the astigmatic aberration as a 4 th astigmatic aberration function;
a function of 1 time 2 of the 4 th astigmatic aberration function is used as an astigmatic aberration function specific to the lens to be inspected which represents an astigmatic aberration component specific to the lens to be inspected, and the astigmatic aberration component specific to the lens to be inspected is obtained from the astigmatic aberration function specific to the lens to be inspected.
5. The systematic error correction method of interferometer apparatus of claim 2,
classifying a function in which a difference between the 1 st coma aberration function and the 2 nd coma aberration function is obtained into aberration functions corresponding to respective seidel aberrations, and determining an aberration function corresponding to a coma aberration as a 4 th coma aberration function;
a function of 1 time 2 of the 4 th coma aberration function is used as a coma aberration function specific to the lens to be detected representing the coma aberration component specific to the lens to be detected, and the coma aberration component specific to the lens to be detected is obtained from the coma aberration function specific to the lens to be detected.
6. The systematic error correction method of interferometer apparatus of claim 3,
classifying a function in which a difference between the 1 st astigmatic aberration function and the 2 nd astigmatic aberration function is obtained into aberration functions corresponding to respective seidel aberrations, and determining an aberration function corresponding to the astigmatic aberration as a 4 th astigmatic aberration function; classifying a function in which a difference between the 1 st coma aberration function and the 2 nd coma aberration function is obtained into aberration functions corresponding to respective seidel aberrations, and determining an aberration function corresponding to a coma aberration as a 4 th coma aberration function;
determining an astigmatic aberration component specific to the lens to be inspected from a function of 1-fold of 2-fold of the 4 th astigmatic aberration function as an astigmatic aberration function specific to the lens to be inspected, the astigmatic aberration function specific to the lens to be inspected representing an astigmatic aberration component specific to the lens to be inspected; a function of 1 time 2 of the 4 th coma aberration function is used as a coma aberration function specific to the lens to be detected, which represents the coma aberration component specific to the lens to be detected, and the coma aberration component specific to the lens to be detected is obtained from the coma aberration function specific to the lens to be detected.
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